In fact, the link between the \(U^3\)  inverse theorem and inverse sumset results runs deeper. It was shown in [1]}, [2]} that a \(U^3\)  inverse theorem with (to date conjectural) polynomial bounds is equivalent to the polynomial Freiman-Ruzsa conjecture, one of the central open problems in additive combinatorics.
Given this, one cannot help but wonder whether there is a more general inverse sumset phenomenon that would naturally correspond to quantitative inverse theorems for the \(U^k\)  norms. In a recent breakthrough, Gowers and Milićević [3]} showed that this is indeed the case, at least for the \(U^4\)  norm. They used a bilinear generalization of theorem:sanders to obtain a quantitative \(U^4\)  inverse theorem.
